March  2015, 20(2): 587-597. doi: 10.3934/dcdsb.2015.20.587

Concentration phenomenon in a nonlocal equation modeling phytoplankton growth

1. 

Department of Mathematics, Henan Normal University, Xinxiang, Henan 453007, China

2. 

Department of Mathematics, Hebei University of Engineering, Handan, Hebei 056021, China, China

Received  June 2013 Revised  November 2014 Published  January 2015

We study a nonlocal reaction-diffusion-advection equation arising from the study of a single phytoplankton species competing for light in a poorly mixed water column. When the diffusion coefficient is very small, the phytoplankton population concentrates around certain zeros of the advection function. The corresponding phytoplankton distribution approaches a $\delta$-like function centered at those zeros.
Citation: Linfeng Mei, Wei Dong, Changhe Guo. Concentration phenomenon in a nonlocal equation modeling phytoplankton growth. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 587-597. doi: 10.3934/dcdsb.2015.20.587
References:
[1]

F. Belgacem, Elliptic Boundary Value Problems with Indefinite Weights: Variational Formulations of the Principal Eigenvalue and Applications,, Pitman Res, (1997).   Google Scholar

[2]

X. Chen and Y. Lou, Principal eigenvalue and eigenfunctions of an elliptic operator with large advection and its application to a competition model,, Indiana Univ. Math. J., 57 (2008), 627.  doi: 10.1512/iumj.2008.57.3204.  Google Scholar

[3]

Y. Du and S.-B. Hsu, Concentration phenomena in a nonlocal quasi-linear problem modelling phytoplankton I: Existence,, SIAM J. Math. Anal., 40 (2008), 1419.  doi: 10.1137/07070663X.  Google Scholar

[4]

Y. Du and S.-B. Hsu, Concentration phenomena in a nonlocal quasi-linear problem modelling phytoplankton II: Limiting profile,, SIAM J. Math. Anal., 40 (2008), 1441.  doi: 10.1137/070706641.  Google Scholar

[5]

Y. Du and S.-B. Hsu, On a nonlocal reaction-diffusion problem arising from the modelling of phytoplankton growth,, SIAM J. Math. Anal., 42 (2010), 1305.  doi: 10.1137/090775105.  Google Scholar

[6]

Y. Du and L. Mei, On a nonlocal reaction-diffusion-advection equation modeling phytoplankton dynamics,, Nonlinearity, 24 (2011), 319.  doi: 10.1088/0951-7715/24/1/016.  Google Scholar

[7]

W. M. Durham, J. O. Kessler and R. Stoker, Disruptive of vertical motility by shear triggers formation of phytoplankton layers,, Science, 323 (2009), 1067.  doi: 10.1126/science.1167334.  Google Scholar

[8]

J. Huisman, M. Arrayas, U. Ebert and B.Sommeijer, How do sinking phytoplankton species manage to persist?, Amer. Naturalist, 159 (2002), 245.  doi: 10.1086/338511.  Google Scholar

[9]

U. Ebert, M. Arrays, N. Temme, B. Sommeijer and J. Huisman, Critical conditions for phytoplankton blooms,, Bull. Math. Biol., 63 (2001), 1095.  doi: 10.1006/bulm.2001.0261.  Google Scholar

[10]

J. Huisman, P. Oostveen and F. J. Weissing, Species dynamics in phytoplankton blooms: Incomplete mixing and competition for light,, Amer. Naturalist, 154 (1999), 46.  doi: 10.1086/303220.  Google Scholar

[11]

J. Huisman, N. N. Pham Thi, D. K. Karl and B. Sommeijer, Reduced mixing generates oscillations and chaos in oceanic deep chlorophyll,, Nature, 439 (2006), 322.  doi: 10.1038/nature04245.  Google Scholar

[12]

J. Huisman and F. J. Wessing, Light-limited growth and competition for light in well-mixed aquatic environments: An elementary model,, Ecology, 75 (1994), 507.  doi: 10.2307/1939554.  Google Scholar

[13]

J. Huisman and F. J. Wessing, Competition for nutrients and light in a mixed water column: A theoretical analysis,, Amer. Naturalist, 146 (1995), 536.  doi: 10.1086/285814.  Google Scholar

[14]

S.-B. Hsu and Y. Lou, Single phytoplankton species growth with light and advection in a water column,, SIAM J. Appl. Math., 70 (2010), 2942.  doi: 10.1137/100782358.  Google Scholar

[15]

H. Ishii and I. Takagi, Global stability of stationary solutions to a nonlinear diffusion equation in phytoplankton dynamics,, J. Math. Biology, 16 (1982), 1.  doi: 10.1007/BF00275157.  Google Scholar

[16]

C. A. Klausmeier and E. Litchman, Algae games: The vertical distribution of pytoplankton in poorly mixed water columns,, Limnol. Oceanogr., 46 (2001), 1998.   Google Scholar

[17]

C. A. Klausmeier, E. Litchman and S. A. Levin, Phytoplankton growth and stoichimetry under multiple nutrient limitation,, Limnol. Oceanogr., 49 (2004), 1463.  doi: 10.4319/lo.2004.49.4_part_2.1463.  Google Scholar

[18]

E. Litchman, C. A. Klausmeier, J. R. Miller, O. M. Schofield and P. G. Falkowski, Multi-nutrient, multi-group model of present and future oceanic phytoplankton communities,, Biogeosci., 3 (2006), 585.  doi: 10.5194/bg-3-585-2006.  Google Scholar

[19]

N. Shigesada and A. Okubo, Analysis of the self-shading effect on algal vertical distribution in natural waters,, J. Math. Biol., 12 (1981), 311.  doi: 10.1007/BF00276919.  Google Scholar

[20]

K. Yoshiyama, J. P. Mellard, E. Litchman and C. A. Klausmeier, Phytoplankton competition for nutrients and light in a stratified water column,, Amer. Naturalist, 174 (2009), 190.  doi: 10.1086/600113.  Google Scholar

[21]

F. J. Weissing and J. Huisman, Growth and competition in a lighted gradient,, J. Theoret. Biol., 168 (1994), 323.  doi: 10.1006/jtbi.1994.1113.  Google Scholar

show all references

References:
[1]

F. Belgacem, Elliptic Boundary Value Problems with Indefinite Weights: Variational Formulations of the Principal Eigenvalue and Applications,, Pitman Res, (1997).   Google Scholar

[2]

X. Chen and Y. Lou, Principal eigenvalue and eigenfunctions of an elliptic operator with large advection and its application to a competition model,, Indiana Univ. Math. J., 57 (2008), 627.  doi: 10.1512/iumj.2008.57.3204.  Google Scholar

[3]

Y. Du and S.-B. Hsu, Concentration phenomena in a nonlocal quasi-linear problem modelling phytoplankton I: Existence,, SIAM J. Math. Anal., 40 (2008), 1419.  doi: 10.1137/07070663X.  Google Scholar

[4]

Y. Du and S.-B. Hsu, Concentration phenomena in a nonlocal quasi-linear problem modelling phytoplankton II: Limiting profile,, SIAM J. Math. Anal., 40 (2008), 1441.  doi: 10.1137/070706641.  Google Scholar

[5]

Y. Du and S.-B. Hsu, On a nonlocal reaction-diffusion problem arising from the modelling of phytoplankton growth,, SIAM J. Math. Anal., 42 (2010), 1305.  doi: 10.1137/090775105.  Google Scholar

[6]

Y. Du and L. Mei, On a nonlocal reaction-diffusion-advection equation modeling phytoplankton dynamics,, Nonlinearity, 24 (2011), 319.  doi: 10.1088/0951-7715/24/1/016.  Google Scholar

[7]

W. M. Durham, J. O. Kessler and R. Stoker, Disruptive of vertical motility by shear triggers formation of phytoplankton layers,, Science, 323 (2009), 1067.  doi: 10.1126/science.1167334.  Google Scholar

[8]

J. Huisman, M. Arrayas, U. Ebert and B.Sommeijer, How do sinking phytoplankton species manage to persist?, Amer. Naturalist, 159 (2002), 245.  doi: 10.1086/338511.  Google Scholar

[9]

U. Ebert, M. Arrays, N. Temme, B. Sommeijer and J. Huisman, Critical conditions for phytoplankton blooms,, Bull. Math. Biol., 63 (2001), 1095.  doi: 10.1006/bulm.2001.0261.  Google Scholar

[10]

J. Huisman, P. Oostveen and F. J. Weissing, Species dynamics in phytoplankton blooms: Incomplete mixing and competition for light,, Amer. Naturalist, 154 (1999), 46.  doi: 10.1086/303220.  Google Scholar

[11]

J. Huisman, N. N. Pham Thi, D. K. Karl and B. Sommeijer, Reduced mixing generates oscillations and chaos in oceanic deep chlorophyll,, Nature, 439 (2006), 322.  doi: 10.1038/nature04245.  Google Scholar

[12]

J. Huisman and F. J. Wessing, Light-limited growth and competition for light in well-mixed aquatic environments: An elementary model,, Ecology, 75 (1994), 507.  doi: 10.2307/1939554.  Google Scholar

[13]

J. Huisman and F. J. Wessing, Competition for nutrients and light in a mixed water column: A theoretical analysis,, Amer. Naturalist, 146 (1995), 536.  doi: 10.1086/285814.  Google Scholar

[14]

S.-B. Hsu and Y. Lou, Single phytoplankton species growth with light and advection in a water column,, SIAM J. Appl. Math., 70 (2010), 2942.  doi: 10.1137/100782358.  Google Scholar

[15]

H. Ishii and I. Takagi, Global stability of stationary solutions to a nonlinear diffusion equation in phytoplankton dynamics,, J. Math. Biology, 16 (1982), 1.  doi: 10.1007/BF00275157.  Google Scholar

[16]

C. A. Klausmeier and E. Litchman, Algae games: The vertical distribution of pytoplankton in poorly mixed water columns,, Limnol. Oceanogr., 46 (2001), 1998.   Google Scholar

[17]

C. A. Klausmeier, E. Litchman and S. A. Levin, Phytoplankton growth and stoichimetry under multiple nutrient limitation,, Limnol. Oceanogr., 49 (2004), 1463.  doi: 10.4319/lo.2004.49.4_part_2.1463.  Google Scholar

[18]

E. Litchman, C. A. Klausmeier, J. R. Miller, O. M. Schofield and P. G. Falkowski, Multi-nutrient, multi-group model of present and future oceanic phytoplankton communities,, Biogeosci., 3 (2006), 585.  doi: 10.5194/bg-3-585-2006.  Google Scholar

[19]

N. Shigesada and A. Okubo, Analysis of the self-shading effect on algal vertical distribution in natural waters,, J. Math. Biol., 12 (1981), 311.  doi: 10.1007/BF00276919.  Google Scholar

[20]

K. Yoshiyama, J. P. Mellard, E. Litchman and C. A. Klausmeier, Phytoplankton competition for nutrients and light in a stratified water column,, Amer. Naturalist, 174 (2009), 190.  doi: 10.1086/600113.  Google Scholar

[21]

F. J. Weissing and J. Huisman, Growth and competition in a lighted gradient,, J. Theoret. Biol., 168 (1994), 323.  doi: 10.1006/jtbi.1994.1113.  Google Scholar

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